# Variational Multiscale Closures for Finite Element Discretizations Using the Mori-Zwanzig Approach

## TL;DR

This paper introduces a novel coarse-grained modeling approach combining Variational Multiscale decomposition with the Mori-Zwanzig formalism to improve finite element simulations of multiscale problems, adaptively determining memory effects.

## Contribution

It develops a parameter-free, adaptive sub-scale model for multiscale finite element discretizations using Mori-Zwanzig formalism, applicable to turbulence and Burgers equation.

## Key findings

- Model effectively captures unresolved scale effects.
- Adaptive memory length improves simulation accuracy.
- Applicable to turbulence and nonlinear problems.

## Abstract

Simulation of multiscale problems remains a challenge due to the disparate range of spatial and temporal scales and the complex interaction between the resolved and unresolved scales. This work develops a coarse-grained modeling approach for the Continuous Galerkin discretizations by combining the Variational Multiscale decomposition and the Mori-Zwanzig (M-Z) formalism. An appeal of the M-Z formalism is that - akin to Greens functions for linear problems - the impact of unresolved dynamics on resolved scales can be formally represented as a convolution (or memory) integral in a non-linear setting. To ensure tractable and efficient models, Markovian closures are developed for the M-Z memory integral. The resulting sub-scale model has some similarities to adjoint stabilization and orthogonal subscale models. The model is made parameter free by adaptively determining the memory length during the simulation. To illustrate the generalizablity of this model, it is employed in coarse-grained simulations for the one-dimensional Burgers equation and in incompressible turbulence problems.